Properties

Label 2240.2.k.a
Level $2240$
Weight $2$
Character orbit 2240.k
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{3} q^{5} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -\zeta_{12}^{3} q^{5} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( 1 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} + ( -1 + 2 \zeta_{12}^{2} ) q^{15} + ( 2 - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{17} -6 q^{19} + ( -1 - 4 \zeta_{12}^{2} ) q^{21} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} - q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -1 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{29} -6 q^{31} + ( -2 + 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{33} + ( 3 - 2 \zeta_{12}^{2} ) q^{35} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{37} + ( 3 - 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{39} + ( 2 - 4 \zeta_{12}^{2} ) q^{41} -2 \zeta_{12}^{3} q^{43} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} + ( 3 - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{51} -2 q^{53} + ( -2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{57} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{59} + ( 2 - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{61} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( 2 - 4 \zeta_{12}^{2} ) q^{67} + ( -2 + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{69} + ( 2 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{71} + ( -4 + 8 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{73} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{75} + ( 6 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{77} + ( 5 - 10 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{79} -9 q^{81} + ( -12 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{83} + ( 3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{85} + ( -12 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{87} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{89} + ( -9 - 8 \zeta_{12} + 6 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{91} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{93} + 6 \zeta_{12}^{3} q^{95} + ( 6 - 12 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{19} - 12q^{21} - 4q^{25} - 4q^{29} - 24q^{31} + 8q^{35} + 24q^{37} - 4q^{49} - 8q^{53} - 8q^{55} + 12q^{65} + 16q^{77} - 36q^{81} - 48q^{83} + 12q^{85} - 48q^{87} - 24q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1791.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0 −1.73205 0 1.00000i 0 1.73205 + 2.00000i 0 0 0
1791.2 0 −1.73205 0 1.00000i 0 1.73205 2.00000i 0 0 0
1791.3 0 1.73205 0 1.00000i 0 −1.73205 + 2.00000i 0 0 0
1791.4 0 1.73205 0 1.00000i 0 −1.73205 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.k.a 4
4.b odd 2 1 2240.2.k.b 4
7.b odd 2 1 2240.2.k.b 4
8.b even 2 1 140.2.g.a 4
8.d odd 2 1 140.2.g.b yes 4
24.f even 2 1 1260.2.c.b 4
24.h odd 2 1 1260.2.c.a 4
28.d even 2 1 inner 2240.2.k.a 4
40.e odd 2 1 700.2.g.g 4
40.f even 2 1 700.2.g.f 4
40.i odd 4 1 700.2.c.b 4
40.i odd 4 1 700.2.c.e 4
40.k even 4 1 700.2.c.c 4
40.k even 4 1 700.2.c.f 4
56.e even 2 1 140.2.g.a 4
56.h odd 2 1 140.2.g.b yes 4
56.j odd 6 1 980.2.o.b 4
56.j odd 6 1 980.2.o.d 4
56.k odd 6 1 980.2.o.b 4
56.k odd 6 1 980.2.o.d 4
56.m even 6 1 980.2.o.a 4
56.m even 6 1 980.2.o.c 4
56.p even 6 1 980.2.o.a 4
56.p even 6 1 980.2.o.c 4
168.e odd 2 1 1260.2.c.a 4
168.i even 2 1 1260.2.c.b 4
280.c odd 2 1 700.2.g.g 4
280.n even 2 1 700.2.g.f 4
280.s even 4 1 700.2.c.c 4
280.s even 4 1 700.2.c.f 4
280.y odd 4 1 700.2.c.b 4
280.y odd 4 1 700.2.c.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 8.b even 2 1
140.2.g.a 4 56.e even 2 1
140.2.g.b yes 4 8.d odd 2 1
140.2.g.b yes 4 56.h odd 2 1
700.2.c.b 4 40.i odd 4 1
700.2.c.b 4 280.y odd 4 1
700.2.c.c 4 40.k even 4 1
700.2.c.c 4 280.s even 4 1
700.2.c.e 4 40.i odd 4 1
700.2.c.e 4 280.y odd 4 1
700.2.c.f 4 40.k even 4 1
700.2.c.f 4 280.s even 4 1
700.2.g.f 4 40.f even 2 1
700.2.g.f 4 280.n even 2 1
700.2.g.g 4 40.e odd 2 1
700.2.g.g 4 280.c odd 2 1
980.2.o.a 4 56.m even 6 1
980.2.o.a 4 56.p even 6 1
980.2.o.b 4 56.j odd 6 1
980.2.o.b 4 56.k odd 6 1
980.2.o.c 4 56.m even 6 1
980.2.o.c 4 56.p even 6 1
980.2.o.d 4 56.j odd 6 1
980.2.o.d 4 56.k odd 6 1
1260.2.c.a 4 24.h odd 2 1
1260.2.c.a 4 168.e odd 2 1
1260.2.c.b 4 24.f even 2 1
1260.2.c.b 4 168.i even 2 1
2240.2.k.a 4 1.a even 1 1 trivial
2240.2.k.a 4 28.d even 2 1 inner
2240.2.k.b 4 4.b odd 2 1
2240.2.k.b 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} - 3 \)
\( T_{19} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( 1 + 14 T^{2} + T^{4} \)
$13$ \( 9 + 42 T^{2} + T^{4} \)
$17$ \( 9 + 42 T^{2} + T^{4} \)
$19$ \( ( 6 + T )^{4} \)
$23$ \( 64 + 32 T^{2} + T^{4} \)
$29$ \( ( -47 + 2 T + T^{2} )^{2} \)
$31$ \( ( 6 + T )^{4} \)
$37$ \( ( 24 - 12 T + T^{2} )^{2} \)
$41$ \( ( 12 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( -3 + T^{2} )^{2} \)
$53$ \( ( 2 + T )^{4} \)
$59$ \( ( -12 + T^{2} )^{2} \)
$61$ \( 576 + 96 T^{2} + T^{4} \)
$67$ \( ( 12 + T^{2} )^{2} \)
$71$ \( 16 + 56 T^{2} + T^{4} \)
$73$ \( 144 + 168 T^{2} + T^{4} \)
$79$ \( 1521 + 222 T^{2} + T^{4} \)
$83$ \( ( 132 + 24 T + T^{2} )^{2} \)
$89$ \( 576 + 96 T^{2} + T^{4} \)
$97$ \( 9801 + 234 T^{2} + T^{4} \)
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